A set \(\mathcal{B}\) is a basis for a topology iff

• \(\mathcal{B}\) is closed under intersections,
• Every \(x\in X\) is in some basic set,
• If \(x\) is in the intersection of two basis sets \(B_1 \cap B_2\), there is a third basic open \(B_3 \ni x\) with \(B_3 \subset B_1 \cap B_2\).

The topology generated by \(\mathcal{B}\) is the following: \(U\subseteq X\) is open iff for each \(x\in U\) there is a basic open \(B\) with \(x\in B \subset U\). Equivalently, every open set is a union of basic open sets.

A set \(S\) in a metric space \((X, d)\) is bounded iff there exists an \(m\in {\mathbb{R}}\) such that \(d(x, y) < m\) for every \(x, y\in S\).

Given two topologies \(\tau_1, \tau_2\),

• \(\tau_1\) is finer than \(\tau_2\) iff \(\tau_1 \supseteq\tau_2\).
• \(\tau_1\) is coarser than \(\tau_2\) iff \(\tau_1 \supseteq\tau_2\).

Two topologies are comparable if either \(\tau_1 < \tau_2\) or \(\tau_2 < \tau_1\).

Note: more open sets is like having a “finer” resolution.

A space \(X\) is connected iff there does not exist a disconnection \(X = A{\coprod}B\) with \(A, B\) nonempty open sets. I.e. \(X\) can not be written as the disjoint union of two proper nonempty open sets. Equivalently, \(X\) contains no proper nonempty clopen sets.

Note that there is an additional condition for a subspace \(Y\subset X\) to be connected: \begin{align*} { \operatorname{cl}} _{Y}(A) \cap V = A \cap{ \operatorname{cl}} _{Y}(B) = \emptyset .\end{align*}

Set \(x\sim y\) iff there exists a connected set \(U\ni x, y\) and take equivalence classes. These classes are the connected components of \(X\).

• A set is closed if and only if its complement is open.
• A set is closed iff it contains all of its limit points.
• A closed set in a subspace: \(Y\subset X \implies { \operatorname{cl}} _{Y}(A) \mathrel{\vcenter{:}}={ \operatorname{cl}} _{X}(A)\cap Y\).

See .

For \(U \subseteq X\), the closure of \(U\) in \(X\) is given by \({ \operatorname{Cl}} _X(U) = \cap_{\substack{ B\supseteq U \\ \text{ closed} }} B\), the intersection of all closed sets in \(X\) containing \(U\). For \(Y\subseteq X\) a subspace containing \(U\), the closure of \(U\) in \(Y\) is \({ \operatorname{Cl}} _Y(U) = { \operatorname{Cl}} _X(U) \cap Y\).¹ In general, we write \(\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu \mathrel{\vcenter{:}}={ \operatorname{cl}} _X(U)\).

An equivalent condition: \(x\in \mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu \iff\) every neighborhood of \(x\) intersects \(U\).²

A topological space \((X, \tau)\) is compact iff every open cover has a finite subcover. That is, if \(\left\{{U_{j}}\right\}_{j\in J} \subseteq \tau\) is a collection of open sets such that \(X = \displaystyle\bigcup_{j\in J} U_{j}\), then there exists a finite subset \(J' \subset J\) such that \(X \subseteq \displaystyle\bigcup_{j\in J'} U_{j}\).

A map \(f:X\to Y\) between topological spaces is continuous if and only if whenever \(U \subseteq Y\) is open, \(f ^{-1} (U) \subseteq X\) is open.

A collection of subsets \(\left\{{U_\alpha}\right\}\) of \(X\) is said to cover \(X\) iff \(X = \cup_{\alpha} U_\alpha\). If \(A\subseteq X\) is a subspace, then this collection covers \(A\) iff \(A\subseteq \cup_{\alpha} U_\alpha\).

A subspace \(Q\subset X\) is dense iff every neighborhood of every point in \(x\) intersects \(Q\). Equivalently, \({ \operatorname{cl}} _X(Q) = Q\).

A space is first-countable iff every point admits a countable neighborhood basis.

A topological space \(X\) is Hausdorff iff points can be separated by disjoint neighborhoods: for every \(p\neq q \in X\) there exist disjoint open sets \(U\ni p\) and \(V\ni q\).

A map \(\iota:A\to B\) with a left inverse \(f:B\to A\) satisfying \(f\circ \iota = \operatorname{id}_A\). Note that this is equivalent to \(f(x) = f(y) \implies x = y\).

For \((X, d)\) a compact metric space and \(\left\{{U_\alpha}\right\}\rightrightarrows X\), there exists a Lebesgue number \(\delta_{L} > 0\) which satisfies \begin{align*} A\subset X, ~ {\operatorname{diam}}(A) < \delta_{L} \implies A\subseteq U_\alpha \text{ for some } \alpha .\end{align*}

For \(A\subset X\), \(x\) is a limit point of \(A\) if every punctured neighborhood \(P_{x}\) of \(x\) intersects \(A\). I.e., every neighborhood of \(x\) intersects \(A\) at a point other than \(x\). Equivalently, \(x\in { \operatorname{cl}} _{X}(A\setminus\left\{{x}\right\})\).

A space is locally connected iff every neighborhood of every point admits a smaller connected neighborhood. I.e. for all \(x\in X\), for all \(N_x \ni x\), there exists a connected set \(U \subset X\) with \(x\in U\).

A space \(X\) is locally compact iff every \(x\in X\) has a neighborhood contained in a compact subset of \(X\).

Note: authors such as Hartshorne often also require that \(X\) is Hausdorff, and refer to the above definition as quasicompactness.

A collection of subsets \({\mathcal{S}}\) of \(X\) is locally finite iff each point of \(M\) has a neighborhood that intersects at most finitely many elements of \({\mathcal{S}}\).

A space \(X\) is locally path-connected iff every point in \(X\) admits some path-connected neighborhood. Equivalently, \(X\) admits a basis of path-connected open subsets.

A neighborhood of a point \(x\) is any open set containing \(x\).

A space is normal if any two disjoint closed subsets can be separated by neighborhoods.

If \(p\in X\), a neighborhood basis at \(p\) is a collection \({\mathcal{B}}_{p}\) of neighborhoods of \(p\) such that if \(N_{p}\) is a neighborhood of \(p\), then \(N_{p} \supseteq B\) for at least one \(B\in {\mathcal{B}}_{p}\).

A map \(f:X\to Y\) is an open map (respectively a closed map) if and only if whenever \(U \subseteq X\) is open (resp. closed), \(f(U)\) is again open (resp. closed)>

A topological space \(X\) is paracompact iff every open cover of \(X\) admits an open locally finite refinement.

A map \(q:X\to Y\) is a quotient map if and only if

1. \(q\) is surjective, and
2. \(U \subseteq Y\) is open if and only if \(q ^{-1} (U)\) is open.

A space \(X\) is path connected if and only if for every pair of points \(x\neq y\) there exists a continuous map \(f:I \to X\) such that \(f(0) = x\) and \(f(1) = y\).

Set \(x\sim y\) iff there exists a path-connected set \(U\ni x, y\) and take equivalence classes.

A subset \(A\subseteq X\) is precompact iff \({ \operatorname{cl}} _{X}(A)\) is compact.

For \((X, \tau_X)\) and \((Y, \tau_Y)\) topological spaces, defining \begin{align*} \tau_{X \times Y} \mathrel{\vcenter{:}}=\left\{{U \times V {~\mathrel{\Big|}~}U\in \tau_X,\, V\in \tau_Y}\right\} \end{align*} yields the product topology on \(X \times Y\).

A topological space \(X\) (possible non-Hausdorff) is quasi-compact iff every open cover admits a finite subcover. If \(X\) is additionally Hausdorff, \(X\) is said to be compact.

Note: this is a distinction coming from algebraic geometry, and Hartshorne in particular.

A cover \({\mathcal{V}}\rightrightarrows X\) is a refinement of \({\mathcal{U}}\rightrightarrows X\) iff for each \(V\in {\mathcal{V}}\) there exists a \(U\in{\mathcal{U}}\) such that \(V\subseteq U\).

A space \(X\) is regular if whenever \(x\in X\) and \(F\not\ni x\) is closed, \(F\) and \(x\) are separated by neighborhoods.

A map \(r\) in \(A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} B\) satisfying \begin{align*}r\circ\iota = \operatorname{id}_{A}.\end{align*} A retract of \(B\) onto a subspace \(A\) is a map \(r:B\to A\) that is a left-inverse for the inclusion \(f:A\hookrightarrow B\), so \(r \circ f = \operatorname{id}_A\):

Link to (partial) Diagram

Equivalently, a continuous map \(r:B\to A\) with \({ \left.{{r}} \right|_{{A}} } = \operatorname{id}_A\) restricting to the identity on \(A\), i.e. fixing \(A\) pointwise. Note that \(r\) is necessarily a surjection.

Alt: Let \(X\) be a topological space and \(A \subset X\) be a subspace, then a retraction of \(X\) onto \(A\) is a map \(r: X\to X\) such that the image of \(X\) is \(A\) and \(r\) restricted to \(A\) is the identity.

A subset \(U \subseteq X\) is saturated with respect to a surjective map \(p: X\twoheadrightarrow Y\) if and only if whenever \(U \cap p ^{-1} (y) = V \neq \emptyset\), we have \(V \subseteq U\), i.e. \(U\) contains every set \(p ^{-1} (y)\) that it intersects. Equivalently, \(U\) is the complete inverse image of a subset of \(Y\).

A space \(X\) is separable iff \(X\) contains a countable dense subset.

A space is second-countable iff it admits a countable basis.

For \((X, \tau)\) a topological space and \(U \subseteq X\) an arbitrary subset, the space \((U, \tau_U)\) is a topological space with a subspace topology defined by \begin{align*} \tau_U \mathrel{\vcenter{:}}=\left\{{Y \cap U {~\mathrel{\Big|}~}U \in \tau}\right\} .\end{align*}

A map \(\pi\) with a right inverse \(f\) satisfying \begin{align*}\pi \circ f = \operatorname{id}\end{align*}

• \(T_0\): Points are distinguishable. For any 2 points \(x_1\neq x_2\), at least one \(x_i\) (say \(x_1\)) admits a neighborhood not containing \(x_2\).

• \(T_1\): For any 2 points, both admit neighborhoods not containing the other. Equivalently, points are closed.

• \(T_2\): For any 2 points, both admit disjoint separating neighborhoods.

• \(T_{2.5}\): For any 2 points, both admit disjoint closed separating neighborhoods.

• \(T_3\): \(T_0\) & regular. Given any point \(x\) and any closed \(F\not\ni x\), there are neighborhoods separating \(F\) and \(x\).

• \(T_{3.5}\): \(T_0\) & completely regular. Any point \(x\) and closed \(F\not\ni x\) can be separated by a continuous function.

• \(T_4\): \(T_1\) & normal. Any two disjoint closed subsets can be separated by neighborhoods.

• Using open sets: closed under arbitrary unions and finite intersections.
• Using closed sets: closed under arbitrary intersections and finite unions.

A topological embedding is a continuous map \(f:X\to Y\) which is a homeomorphism onto its image, i.e. \(X\cong_{{\mathsf{Top}}} f(X)\).

For \(f: (X, d_{x}) \to (Y, d_{Y})\) metric spaces, \begin{align*} \forall \varepsilon> 0, ~\exists \delta > 0 \text{ such that } \quad d_{X}(x_{1}, x_{2}) < \delta \implies d_{Y}(f(x_{1}), f(x_{2})) < \varepsilon .\end{align*}

For an \(R{\hbox{-}}\)module \(M\), a basis \(B\) is a linearly independent generating set.

Points \(x\in M^n\) defined by \begin{align*} {\partial}M = \left\{{x\in M: H_{n}(M, M-\left\{{x}\right\}; {\mathbb{Z}}) = 0}\right\} \end{align*}

Denoting \(\Delta^p \xrightarrow{\sigma} X \in C_{p}(X; G)\), a map that sends pairs (\(p{\hbox{-}}\)chains, \(q{\hbox{-}}\)cochains) to \((p-q){\hbox{-}}\)chains \(\Delta^{p-q} \to X\) by \begin{align*} H_{p}(X; R)\times H^q(X; R) \xrightarrow{\frown} H_{p-q}(X; R)\\ \sigma \frown \psi = \psi(F_{0}^q(\sigma))F_{q}^p(\sigma) \end{align*} where \(F_{i}^j\) is the face operator, which acts on a simplicial map \(\sigma\) by restriction to the face spanned by \([v_{i} \ldots v_{j}]\), i.e. \(F_{i}^j(\sigma) = {\left.{{\sigma}} \right|_{{[v_{i} \ldots v_{j}]}} }\).

A map \(X \xrightarrow{f} Y\) is said to be cellular if \(f(X^{(n)}) \subseteq Y^{(n)}\) where \(X^{(n)}\) denotes the \(n{\hbox{-}}\) skeleton.

An element \(c \in C_{p}(X; R)\) can be represented as the singular \(p\) simplex \(\Delta^p \to X\).

Given two maps between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f, ~g} (D_*, {\partial}_{D})\), a chain homotopy is a family \(h_{i}: C_{i} \to B_{i+1}\) satisfying \begin{align*}f_{i}-g_{i} = {\partial}_{B, i-1}\circ h_{n} + h_{i+1}\circ {\partial}_{A, i}\end{align*} .

A map between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f} (D_*, {\partial}_{D})\) is a chain map iff each component \(C_{i} \xrightarrow{f_{i}} D_{i}\) satisfies \begin{align*} f_{i-1}\circ{\partial}_{C, i} = {\partial}_{D,i} \circ f_{i} \end{align*} (i.e this forms a commuting ladder)

A manifold that is compact, with or without boundary.

An cochain \(c \in C^p(X; R)\) is a map \(c \in \hom(C_{p}(X; R), R)\) on chains.

A constant map \(f: X\to Y\) iff \(f(X) = y_{0}\) for some \(y_{0}\in Y\), i.e. for every \(x\in X\) the output value \(f(x) = y_{0}\) is the same.

For a directed system \((X_{i}, f_{ij}\), the colimit is an object \(X\) with a sequence of projections \(\pi_{i}:X\to X_{i}\) such that for any \(Y\) mapping into the system, the following diagram commutes:

A space \(X\) is contractible if \(\operatorname{id}_X\) is nullhomotopic. i.e. the identity is homotopic to a constant map \(c(x) = x_0\).

Equivalently, \(X\) is contractible if \(X \simeq\left\{{x_0}\right\}\) is homotopy equivalent to a point. This means that there exists a mutually inverse pair of maps \(f: X \to\left\{{x_0}\right\}\) and \(g:\left\{{x_0}\right\} \to X\) such that \(f\circ g \simeq\operatorname{id}_{\left\{{x_0}\right\}}\) and \(g\circ f \simeq\operatorname{id}_X\).³

A covering space of \(X\) is the data \(p: \tilde X \to X\) such that

1. Each \(x\in X\) admits a neighborhood \(U\) such that \(p ^{-1} (U)\) is a union of disjoint open sets in \(\tilde V_i \subseteq X\) (the sheets of \(\tilde X\) over \(U\)),
2. \({ \left.{{p}} \right|_{{V_i}} }: V_i \to U\) is a homeomorphism for each sheet.

An isomorphism of covering spaces \(\tilde X_1 \cong \tilde X_2\) is a commutative diagram

Link to diagram

A map taking pairs (\(p{\hbox{-}}\)cocycles, \(q{\hbox{-}}\)cocycles) to \((p+q){\hbox{-}}\)cocyles by \begin{align*} H^p(X; R) \times H^q(X; R) \xrightarrow{\smile} H^{p+q}(X; R)\\ (a \cup b)(\sigma) = a(\sigma \circ I_{0}^p)~b(\sigma \circ I_{p}^{p+q}) \end{align*} where \(\Delta^{p+q} \xrightarrow{\sigma} X\) is a singular \(p+q\) simplex and

\begin{align*}I_{i}^j: [i, \cdots, j] \hookrightarrow\Delta^{p+q} .\end{align*}

is an embedding of the \((j-i){\hbox{-}}\)simplex into a \((p+q){\hbox{-}}\)simplex.

An \(n{\hbox{-}}\)cell of \(X\), say \(e^n\), is the image of a map \(\Phi: B^n \to X\). That is, \(e^n = \Phi(B^n)\). Attaching an \(n{\hbox{-}}\)cell to \(X\) is equivalent to forming the space \(B^n \coprod_{f} X\) where \(f: {\partial}B^n \to X\).

• A \(0{\hbox{-}}\)cell is a point.
• A \(1{\hbox{-}}\)cell is an interval \([-1, 1] = B^1 \subset {\mathbb{R}}^1\). Attaching requires a map from \(S^0 =\left\{{-1, +1}\right\} \to X\)
• A \(2{\hbox{-}}\)cell is a solid disk \(B^2 \subset {\mathbb{R}}^2\) in the plane. Attaching requires a map \(S^1 \to X\).
• A \(3{\hbox{-}}\)cell is a solid ball \(B^3 \subset {\mathbb{R}}^3\). Attaching requires a map from the sphere \(S^2 \to X\).

For a covering space \(\tilde X \xrightarrow{p} X\), self-isomorphisms \(f:\tilde X \to \tilde X\) of covering spaces are referred to as deck transformations.

A map \(r\) in \(A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X\) that is a retraction (so \(r\circ \iota = \operatorname{id}_{A}\)) that also satisfies \(\iota \circ r \simeq\operatorname{id}_{X}\).

Note that this is equality in one direction, but only homotopy equivalence in the other.

Equivalently, a map \(F:I\times X\to X\) such that \begin{align*} F_{0}(x) &= \operatorname{id}_{X} F_{t}(x)\mathrel{\Big|}_{A} &= \operatorname{id}_{A} F_{1}(X) &= A .\end{align*}

Alt:

A deformation retract is a homotopy \(H:X\times I \to X\) from \(\operatorname{id}_X\) to \(\operatorname{id}_A\) where \({ \left.{{H}} \right|_{{A}} } = \operatorname{id}_A\) fixes \(A\) at all times. \begin{align*} H: X\times I \to X \\ H(x, 0) = \operatorname{id}_X \\ H(x, 1) = \operatorname{id}_A \\ x\in A \implies H(x, t) \in A \quad \forall t \end{align*}

Given any \(f: S^n \to S^n\), there are induced maps on homotopy and homology groups. Taking \(f^*: H^n(S^n) \to H^n(S^n)\) and identifying \(H^n(S^n) \cong {\mathbb{Z}}\), we have \(f^*: {\mathbb{Z}}\to{\mathbb{Z}}\). But homomorphisms of free groups are entirely determined by their action on generators. So if \(f^*(1) = n\), define \(n\) to be the degree of \(f\).

For a functor \(T\) and an \(R{\hbox{-}}\)module \(A\), a left derived functor \((L_{nT})\) is defined as \(h_{n}(TP_{A})\), where \(P_{A}\) is a projective resolution of \(A\).

For \(x\in M\), the only nonvanishing homology group \(H_{i}(M, M - \left\{{x}\right\}; {\mathbb{Z}})\)

A functor \(T\) is right exact if a short exact sequence

\begin{align*}0 \to A \to B \to C \to 0 \end{align*} yields an exact sequence

\begin{align*}\ldots TA \to TB \to TC \to 0 \end{align*} and is left exact if it yields

\begin{align*}0 \to TA \to TB \to TC \to \ldots \end{align*} Thus a functor is exact iff it is both left and right exact, yielding

\begin{align*}0 \to TA \to TB \to TC \to 0 .\end{align*}

An \(R{\hbox{-}}\)module is flat if \(A\otimes_{R} {-}\) is an exact functor.

An action \(G\curvearrowright X\) is properly discontinuous if each \(x\in X\) has a neighborhood \(U\) such that all of the images \(g(U)\) for \(g\in G\) are disjoint, i.e. \(g_1(U) \cap g_2(U) \neq \emptyset \implies g_1 = g_2\). The action is free if there are no fixed points.

Sometimes a slightly weaker condition is used: every point \(x\in X\) has a neighborhood \(U\) such that \(U \cap G(U) \neq \emptyset\) for only finitely many \(G\).

A \({\hbox{-}}\)module \(M\) with a basis \(S = \left\{{s_{i}}\right\}\) of generating elements. Every such module is the image of a unique map \(\mathcal{F}(S) = R^S \twoheadrightarrow M\), and if \(M = \left< S \mathrel{\Big|}\mathcal{R} \right>\) for some set of relations \(\mathcal{R}\), then \(M \cong R^S / \mathcal{R}\).

For a connected, closed, orientable manifold, \([M]\) is a generator of \(H_{n}(M; {\mathbb{Z}}) = {\mathbb{Z}}\).

\(S = \left\{{s_{i}}\right\}\) is a generating set for an \(R{\hbox{-}}\) module \(M\) iff \begin{align*}x\in M \implies x = \sum r_{i} s_{i}\end{align*} for some coefficients \(r_{i} \in R\) (where this sum may be infinite).

Let \(X, Y\) be topological spaces and \(f,g: X \to Y\) continuous maps. Then a homotopy from \(f\) to \(g\) is a continuous function

\(F: X \times I \to Y\)

such that

\(F(x, 0) = f(x)\) and \(F(x,1) = g(x)\)

for all \(x\in X\). If such a homotopy exists, we write \(f\simeq g\). This is an equivalence relation on \(\text{Hom}(X,Y)\), and the set of such classes is denoted \([X,Y] \mathrel{\vcenter{:}}=\hom (X,Y)/\simeq\).

Let \(f: X \to Y\) be a continuous map, then \(f\) is said to be a homotopy equivalence if there exists a continuous map \(g: X \to Y\) such that

\(f\circ g \simeq\operatorname{id}_Y\) and \(g\circ f \simeq\operatorname{id}_X\).

Such a map \(g\) is called a homotopy inverse of \(f\), the pair of maps is a homotopy equivalence.

If such an \(f\) exists, we write \(X \simeq Y\) and say \(X\) and \(Y\) have the same homotopy type, or that they are homotopy equivalent.

Note that homotopy equivalence is strictly weaker than homeomorphic equivalence, i.e., \(X\cong Y\) implies \(X \simeq Y\) but not necessarily the converse.

For a manifold \(M\), a map on homology defined by \begin{align*} H_{\widehat{i}}M \otimes H_{\widehat{j}}M \to H_{\widehat{i+j}}X\\ \alpha\otimes\beta \mapsto \left< \alpha, \beta \right> \end{align*} obtained by conjugating the cup product with Poincaré Duality, i.e. 

\begin{align*}\left< \alpha, \beta \right> = [M] \frown ([\alpha] {}^{ \check{} }\smile [\beta] {}^{ \check{} }) .\end{align*}

Then, if \([A], [B]\) are transversely intersecting submanifolds representing \(\alpha, \beta\), then \begin{align*}\left<\alpha, \beta\right> = [A\cap B]\end{align*} . If \(\widehat{i} = j\) then \(\left< \alpha, \beta \right> \in H_{0} M = {\mathbb{Z}}\) is the signed number of intersection points.

Alt: The pairing obtained from dualizing Poincare Duality to obtain \begin{align*}\mathrm{F}(H_{i} M) \otimes\mathrm{F}(H_{n-i}M) \to {\mathbb{Z}}\end{align*} Computed as an oriented intersection number between two homology classes (perturbed to be transverse).

The nondegenerate bilinear form cohomology induced by the Kronecker Pairing: \begin{align*}I: H^k(M_{n}) \times H^{n-k}(M^n) \to {\mathbb{Z}}\end{align*} where \(n=2k\).

• When \(k\) is odd, \(I\) is skew-symmetric and thus a symplectic form.
• When \(k\) is even (and thus \(n \equiv 0 \pmod 4\)) this is a symmetric form.
• Satisfies \(I(x,y) = (-1)^{k(n-k)} I(y, x)\)

A map pairing a chain with a cochain, given by \begin{align*} H^n(X; R) \times H_{n}(X; R) \to R \\ ([\psi, \alpha]) \mapsto \psi(\alpha) \end{align*} which is a nondegenerate bilinear form.

At a point \(x \in V \subset M\), a generator of \(H_{n}(V, V-\left\{{x}\right\})\). The degree of a map \(S^n \to S^n\) is the sum of its local degrees.

A generating \(S\) for a module \(M\) is linearly independent if \(\sum r_{i} s_{i} = 0_M \implies \forall i,~r_{i} = 0\) where \(s_{i}\in S, r_{i} \in R\).

\(H_{n}(X, X-A; {\mathbb{Z}})\) is the local homology at \(A\), also denoted \(H_{n}(X \mathrel{\Big|}A)\)

At a point \(x\in M^n\), a choice of a generator \(\mu_{x}\) of \(H_{n}(M, M - \left\{{x}\right\}) = {\mathbb{Z}}\).

An \(n{\hbox{-}}\)manifold is a Hausdorff space in which each neighborhood has an open neighborhood homeomorphic to \({\mathbb{R}}^n\).

A manifold in which open neighborhoods may be isomorphic to either \({\mathbb{R}}^n\) or a half-space \(\left\{{\mathbf{x} \in {\mathbb{R}}^n \mathrel{\Big|}x_{i} > 0}\right\}\).

A covering space is normal if and only if for every \(x\in X\) and every pair of lifts \(\tilde x_1, \tilde x_2\), there is a deck transformation \(f\) such that \(f(\tilde x_1) = \tilde x_2\).

A map \(X\xrightarrow{f} Y\) is nullhomotopic if it is homotopic to a constant map \(X \xrightarrow{g} \left\{{y_{0}}\right\}\); that is, there exists a homotopy \begin{align*} F: X\times I &\to Y \\ {\left.{{F}} \right|_{{X\times\left\{{0}\right\}}} } &= f \quad F(x, 0) = f(x) \\ {\left.{{F}} \right|_{{X\times\left\{{1}\right\}}} } &= g \quad F(x, 1) = g(x) = y_{0}\\ .\end{align*}

Alt:

If \(f\) is homotopic to a constant map, say \(f: x \mapsto y_0\) for some fixed \(y_0 \in Y\), then \(f\) is said to be nullhomotopic. In other words, if \(f:X\to Y\) is nullhomotopic, then there exists a homotopy \(H: X\times I \to Y\) such that \(H(x, 0) = f(x)\) and \(H(x, 1) = y_0\).

Note that constant maps (or anything homotopic) induce zero homomorphisms.

For a group action \(G\curvearrowright X\), the orbit space \(X/G\) is defined as \(X/\sim\) where \(\forall x,y\in X, x\sim y \iff \exists g\in G \mathrel{\Big|}g.x = y\).

A manifold for which an orientation exists, see “Orientation of a Manifold.”

For any manifold \(M\), a two sheeted orientable covering space \(\tilde M_{o}\). \(M\) is orientable iff \(\tilde M\) is disconnected. Constructed as \begin{align*} \tilde M = \coprod_{x\in M}\left\{{\mu_{x} \mathrel{\Big|}\mu_{x}~ \text{is a local orientation}}\right\} .\end{align*}

A family of \(\left\{{\mu_{x}}\right\}_{x\in M}\) with local consistency: if \(x,y \in U\) then \(\mu_{x}, \mu_{y}\) are related via a propagation.

Formally, a function \begin{align*}M^n \to \coprod_{x\in M} H(X \mathrel{\Big|}\left\{{x}\right\})\\ x \mapsto \mu_{x}\end{align*} such that \(\forall x \exists N_{x}\) in which \(\forall y\in N_{x}\), the preimage of each \(\mu_{y}\) under the map \(H_{n}(M\mathrel{\Big|}N_{x}) \twoheadrightarrow H_{n}(M\mathrel{\Big|}y)\) is a single generator \(\mu_{N_{x}}\).

TFAE:

• \(M\) is orientable.
• The map \(W: (M, x) \to {\mathbb{Z}}_{2}\) is trivial.
• \(\tilde M_{o} = M \coprod {\mathbb{Z}}_{2}\) (two sheets).
• \(\tilde M_{o}\) is disconnected
• The projection \(\tilde M_{o} \to M\) admits a section.

A pairing alone is an \(R{\hbox{-}}\)bilinear module map, or equivalently a map out of a tensor product since \(p: M\otimes_{R} N \to L\) can be partially applied to yield \(\phi: M \to L^N = \hom_{R}(N, L)\). A pairing is perfect when \(\phi\) is an isomorphism.

For a closed, orientable \(n{\hbox{-}}\)manifold, following map \([M] \frown {-}\) is an isomorphism: \begin{align*} D: H^k(M; R) \to H_{n-k}(M; R) \\ D(\alpha) = [M] \frown \alpha\end{align*}

A space \(X\) is semilocally simply connected if every \(x\in X\) has a neighborhood \(U\) such that \(U\hookrightarrow X\) induces the trivial map \(\pi_1(U;x) \to \pi_1(X, x)\).

Given a simplex \(\sigma = [v_1 \cdots v_n]\), define the face map \begin{align*} {\partial}_i:\Delta^n &\to \Delta^{n-1} \\ \sigma &\mapsto [v_1 \cdots \widehat{v}_i \cdots v_n] \end{align*}

A simplicial complex is a set \(K\) satisfying

1. \(\sigma \in K \implies {\partial}_i\sigma \in K\).

2. \(\sigma,\tau\in K \implies \sigma\cap\tau = \emptyset,~ {\partial}_i\sigma,~\text{or}~{\partial}_i\tau\).

This amounts to saying that any collection of \((n-1)\)-simplices uniquely determines an \(n\)-simplex (or its lack thereof), or that that map \(\Delta^k \to X\) is a continuous injection from the standard simplex in \({\mathbb{R}}^n\).

3. \({\left\lvert {K\cap B_\varepsilon(\sigma)} \right\rvert} < \infty\) for every \(\sigma\in K\), identifying \(\sigma \subseteq {\mathbb{R}}^n\).

For a map \begin{align*}K\xrightarrow{f} L\end{align*} between simplicial complexes, \(f\) is a simplicial map if for any set of vertices \(\left\{{v_{i}}\right\}\) spanning a simplex in \(K\), the set \(\left\{{f(v_{i})}\right\}\) are the vertices of a simplex in \(L\).

A space \(X\) is simply connected if and only if \(X\) is path-connected and every loop \(\gamma : S^1 \xrightarrow{} X\) can be contracted to a point.

Equivalently, there exists a lift \(\widehat{\gamma }: D^2 \xrightarrow{} X\) such that \({ \left.{{\widehat{\gamma}}} \right|_{{{{\partial}}D^2}} } = \gamma\).

Equivalently, for any two paths \(p_1, p_2: I \xrightarrow{} X\) such that \(p_1(0) = p_2(0)\) and \(p_1(1) = p_2(1)\), there exists a homotopy \(F: I^2 \xrightarrow{} X\) such that \({ \left.{{F}} \right|_{{0}} } = p_1,\, { \left.{{F}} \right|_{{0}} } = p_2\).

Equivalently, \(\pi _1 X = 1\) is trivial.

\begin{align*}x \in C_{n}(x) \implies X = \sum_{i} n_{i} \sigma_{i} = \sum_{i} n_{i} (\Delta^n \xrightarrow{\sigma_{i}} X) .\end{align*}

\begin{align*}x \in C^n(x) \implies X = \sum_{i} n_{i} \psi_{i} = \sum_{i} n_{i} (\sigma_{i} \xrightarrow{\psi_{i}} X) .\end{align*}

For an \(R{\hbox{-}}\)module \begin{align*} \operatorname{Tor}_{R}^n({-}, B) = L_{n}({-}\otimes_{R} B) ,\end{align*} where \(L_{n}\) denotes the \(n\)th left derived functor.

For a space \(X\), defined as \begin{align*} CX = \frac{X\times I} {X \times\left\{{0}\right\}} .\end{align*} Example: The cone on the circle \(CS^1\)

Note that the cone embeds \(X\) in a contractible space \(CX\).

Compact represented as \(\Sigma X = CX \coprod_{\operatorname{id}_{X}} CX\), two cones on \(X\) glued along \(X\). Explicitly given by

\begin{align*}\Sigma X = \frac{X\times I}{(X\times\left\{{0}\right\}) \cup(X\times\left\{{1}\right\}) \cup(\left\{{x_{0}}\right\} \times I)} .\end{align*}

Given a pointed space \((X,x_{0})\), we define the fundamental group \(\pi_{1}(X)\) as follows:

• Take the set \begin{align*} L \mathrel{\vcenter{:}}=\left\{{\alpha: S^1\to X \mathrel{\Big|}\alpha(0) = \alpha(1) = x_{0}}\right\} .\end{align*}

• Define an equivalence relation \(\alpha \sim \beta\) iff \(\alpha \simeq\beta\) in \(X\), so there exists a homotopy

\begin{align*} H: &S^1 \times I \to X \\ & \begin{cases} H(s, 0) = \alpha(s)\\ H(s, 1) = \beta(s) , \end{cases} \end{align*} - Check that this relation is

• Symmetric: Follows from considering \(H(s, 1-t)\).

• Reflexive: Take \(H(s, t) = \alpha (s)\) for all \(t\).

• Transitive: Follows from reparameterizing.

• Define \(L/\sim\), which contains elements like \([\alpha]\) and \([\operatorname{id}_{x_{0}}]\), the equivalence classes of loops after quotienting by this relation.

• Define a product structure: for \([\alpha], [\beta] \in L/\sim\), define \([\alpha][\beta] = [\alpha \cdot \beta]\), where we just need to define a product structure on actual loops. Do this by reparameterizing: \begin{align*} (\alpha \cdot \beta )(s) \mathrel{\vcenter{:}}= \begin{cases} \alpha (2s) & s \in [0, 1/2] \\ \beta (2s-1) & s \in [1/2, 1] . \end{cases} \end{align*}

• Check that this map is:

  • Continuous: by the pasting lemma and assumed continuity of \(f, g\).

  • Well-defined: ?

• Check that this is actually a group

  • Identity element: The constant loop \(\operatorname{id}_{x_0}: I\to X\) where \(\operatorname{id}_{x_0}(t) = x_0\) for all \(t\).

  • Inverses: The reverse loop \(\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu(t) \mathrel{\vcenter{:}}=\alpha(1-t)\).

  • Closure: Follows from the fact that start/end points match after composing loops, and reparameterizing.

  • Associativity: Follows from reparameterizing.

The sequences splits when a morphism \(f_2^{-1}: C \to B\) exists. In \(\textbf{Ab}\), this means \(B \cong A \oplus C\), in \(\mathbf{Grp}\) it’s \(B \cong A \rtimes_\phi C\).